According to my teacher, in some situations, Z-buffer algorithm needs the painter's algorithm to render a 3D scene in a 2D image.
I thought Z-buffer algorithm as an extension, an enhancement of painter's one : for me, Z-buffer algorithm can do all what painter's one is able to do, and even more (intersection of polygons and cyclic overlaps).
However, in some situations, Z-buffer algorithm can't work correctly if painter's one isn't used... What are these situations ?
At least one reason from wiki
These and other flaws with the algorithm led to the development of Z-buffer techniques, which can be viewed as a development of the painter's algorithm, by resolving depth conflicts on a pixel-by-pixel basis, reducing the need for a depth-based rendering order. Even in such systems, a variant of the painter's algorithm is sometimes employed. As Z-buffer implementations generally rely on fixed-precision depth-buffer registers implemented in hardware, there is scope for visibility problems due to rounding error. These are overlaps or gaps at joints between polygons. To avoid this, some graphics engine implementations "overrender", drawing the affected edges of both polygons in the order given by painter's algorithm. This means that some pixels are actually drawn twice (as in the full painter's algorithm) but this happens on only small parts of the image and has a negligible performance effect.
The only situation I can imagine is when using transparency, because the Z-Buffer does not work when the transparency method used requires sorting. Imagine an upfront primitive already being drawn and the Z-Buffer being filled already, then primitives drawn later would be skipped. When drawing sorted, you can draw from back to front.
Related
I have a program that visualizes triangular meshes and allows the users to draw on the meshes using a pen. I want to have a "snapping" mode in my system. The snapping mode performs drawing corrections for the user in the sense that the user-drawn lines are snapped to the nearest edge (or the silhouette) of that part of the mesh.
I'm looking for an algorithm that compute the edges visible on the mesh from a given point of view. By edges, I'm referring to the outlines of the shape: corner points and the lines between them (similar to the definition of an edge in computer vision/image processing -- such as Canny edges).
So far I've thought of two approaches for this:
Edge detection: so far I've only found this paper. Their method is understandable, yet the implementation is not trivial (due to tensor computations and some ambiguity in their explanations). The problem with this approach is that it produces "edge strength values" which is a value in the range [0, 1] for every vertex. The value of 1 indicates an edge vertex with a high confidence. This introduces extra thresholding parameters in the system which I'd rather not have. Their output looks like this (range [0, 1] scaled to [0, 65535]):
Rendering or non-photorealistic methods such as the one asked in this question or this paper. They seem to be able to create the silhouette that I'm after as can be seen below:
I'm not a graphics expert and as of yet I don't know whether their methods can be used for computation of the feature lines rather than rendering.
I was wondering if anybody has any ideas about a good algorithm for what I want to do. Since the system is very interactive, the performance is important. The snapping feature does not have to be enabled all the time (therefore, if the method is computationally expensive, some delay in when "snapping enabled" mode is toggled can be tolerated while the algorithm is computing the edges.) Also, if you know of any implementation (preferably open source), I'd be grateful if you could share it with me.
There are two types of edges that you want to detect:
silhouette edges are viewpoint dependent, they correspond to the places where the line of sight tangents the surfaces. With a triangulated model, they are easy to determine, as they are shared by a front-facing triangle and a back-facing one.
"angular" edges are viewpoint independent and formed by a discontinuity in the tangent plane direction. As a triangulated model has itself this kind of discontinuity, there is no exact criterion to find them. Just set a threshold on the angle formed by two triangles. This threshold must be such that smooth patches do not trigger.
By this approach, you will find the wanted edges in 3D.
This is not enough, as part of them are hidden by other surfaces. You have the option of integrating them as edges in the 3D model and letting the rendering engine do its job, or, if you have the courage, to implement an hidden lines removal algorithm. (The wikipedia link is a little terse.)
Since posting the question, something else came into my head. Since 2D edge detection is a very well-studied problem, one way of tackling the problem is performing 2D edge detection on the projection image of the mesh.
In other words, given a specific view of the mesh, one could generate a 2D image. A 2D edge detection algorithm (such as Canny edge detector) could then be run on the 2D image and the results can be back-projected to 3D to determine the silhouettes of the mesh in question. One possible advantage of this is simplicity!
Edit (2017):
Even though I moved away from this, I returned to this problem again for a different purpose. To anybody else looking into this problem: there is a paper that talks about various contours from meshes that's worth reading (the paper is "Suggestive Contours for Conveying Shape" by DeCarlo et al.).
Working implementation of the methods discussed in the paper are available here.
I'm working on a purely continuous physics engine, and I need to choose algorithms for broad and narrow phase collision detection. "Purely continuous" means I never do intersection tests, but instead want to find ways to catch every collision before it happens, and put each into "planned collisions" stack that is ordered by TOI.
Broad Phase
The only continuous broad-phase method I can think of is encasing each body in a circle and testing if each circle will ever overlap another. This seems horribly inefficient however, and lacks any culling.
I have no idea what continuous analogs might exist for today's discrete collision culling methods such as quad-trees either. How might I go about preventing inappropriate and pointless broad test's such as a discrete engine does?
Narrow Phase
I've managed to adapt the narrow SAT to a continuous check rather than discrete, but I'm sure there's other better algorithms out there in papers or sites you guys might have come across.
What various fast or accurate algorithm's do you suggest I use and what are the advantages / disatvantages of each?
Final Note:
I say techniques and not algorithms because I have not yet decided on how I will store different polygons which might be concave, convex, round, or even have holes. I plan to make a decision on this based on what the algorithm requires (for instance if I choose an algorithm that breaks down a polygon into triangles or convex shapes I will simply store the polygon data in this form).
You said circles, so I'm assuming you have 2D objects. You could extend your 2D object (or their bounding shapes) into 3D by adding a time dimension, and then you can use the normal techniques for checking for static collisions among a set of 3D objects.
For example, if you have a circle in (x, y) moving to the right (+x) with constant velocity, then, when you extend that with a time dimension, you have a diagonal cylinder in (x, y, t). By doing intersections between these 3D objects (just treat time as z), you can see if two objects will ever intersect. If point P is a point of intersection, then you know the time of that intersection simply by looking at P.t.
This generalizes into higher dimensions, too, though the math gets hard (for me anyway).
The collision detection might be tricky if objects have complex paths. For example, if your circle is influenced by gravity, then the extruded space-time object is a parabolic sphere sweep rather than a simple cylinder. You could pad the bounding objects a bit and use linear approximations over shorter periods of time and iterate, but I'm not sure if that violates what you mean by continuous.
I am going to assume you want things like gravity or other conservative forces in your simulation. If that's the case the trajectories of your objects are most likely not going to be lines, in which case, just like Adrian pointed out, the math will be somewhat harder. I can't think of a way to avoid checking all possible combinations of curves for collisions, but you can calculate the minimum distance between two curves rather easily, as long as both are solutions to linear systems (or, in general, if you have a closed form solution for the curves). If you know that x1(t) = f(t) and x2(t) = g(t) then what you'll want to
do is calculate the distance ||x1(t) - x2(t)|| and set its derivative to zero. This should be an expression that depends on f(t), g(t) and their derivatives and will give you a time tmin (or maybe a few possible ones) at which you then evaluate the distance and check to see if it is greater or smaller than r1+r2 --- the sum of the radii of the two bounding circles. If it is smaller, then you have a potential collision at that time so you run the narrow phase algorithm.
I've been searching far and wide on the seven internets, and have come to no avail. The closest to what I need seems to be The cutting stock problem, only in 2D (which is disappointing since Wikipedia doesn't provide any directions on how to solve that one). Another look-alike problem would be UV unwrapping. There are solutions there, but only those that you get from add-ons on various 3D software.
Cutting the long talk short - what I want is this: given a rectangle of known width and height, I have to find out how many shapes (polygons) of known sizes (which may be rotated at will) may I fit inside that rectangle.
For example, I could choose a T-shaped piece and in the same rectangle I could pack it both in an efficient way, resulting in 4 shapes per rectangle
as well as tiling them based on their bounding boxes, case in which I could only fit 3
But of course, this is only an example... and I don't think it would be much use to solving on this particular case. The only approaches I can think of right now are either like backtracking in their complexity or solve only particular cases of this problem. So... any ideas?
Anybody up for a game of Tetris (a subset of your problem)?
This is known as the packing problem. Without knowing what kind of shapes you are likely to face ahead of time, it can be very difficult if not impossible to come up with an algorithm that will give you the best answer. More than likely unless your polygons are "nice" polygons (circles, squares, equilateral triangles, etc.) you will probably have to settle for a heuristic that gives you the approximate best solution most of the time.
One general heuristic (though far from optimal depending on the shape of the input polygon) would be to simplify the problem by drawing a rectangle around the polygon so that the rectangle would be just big enough to cover the polygon. (As an example in the diagram below we draw a red rectangle around a blue polygon.)
Once we have done this, we can then take that rectangle and try to fit as many of that rectangle into the large rectangle as possible. This simplfies the problem into a rectangle packing problem which is easier to solve and wrap your head around. An example of an algorithm for this is at the following link:
An Effective Recursive Partitioning Approach for the Packing of Identical Rectangles in a Rectangle.
Now obviously this heuristic is not optimal when the polygon in question is not close to being the same shape as a rectangle, but it does give you a minimum baseline to work with especially if you don't have much knowledge of what your polygon will look like (or there is high variance in what the polygon will look like). Using this algorithm, it would fill up a large rectangle like so:
Here is the same image without the intermediate rectangles:
For the case of these T-shaped polygons, the heuristic is not the best it could be (in fact it may be almost a worst case scenario for this proposed approximation), but it would work very well for other types of polygons.
consider what the other answer said by placing the t's into a square, but instead of just leaving it as a square set the shapes up in a list. Then use True and False to fill the nested list as the shape i.e. [[True,True,True],[False,True,False]] for your T shape. Then use a function to place the shapes on the grid. To optimize the results, create a tracker which will pay attention to how many false in a new shape overlap with trues that are already on the grid from previous shapes. The function will place the shape in the place with the most overlaps. There will have to be modifications to create higher and higher optimizations, but that is the general premise which you are looking for.
I have a set of rectangles and arbitrary shape in 2D space. The shape is not necessary a polygon (it may be a circle), and rectangles have different widths and heights. The task is to approximate the shape with rectangles as close as possible. I can't change rectangles dimensions, but rotation is permitted.
It sounds very similar to packing problem and covering problem but covering area is not rectangular...
I guess it's NP problem, and I'm pretty sure there should be some papers that show good heuristics to solve it, but I don't know what to google? Where should I start?
Update: One idea just came into my mind but I'm not sure if it's worth investigating. What if we consider bounding shape as a physical mold filled with water. Each rectangle is considered as a positively charged particle with size. Now drop the smallest rectangle to it. Then drop the next by size at random point. If rectangles too close they repel each other. Keep adding rectangles until all are used. Could this method work?
I think you could look for packing and automatic layout generation algorithms. Automatic VLSI layout generation algorithms might need similar things, just like textile layout questions...
This paper Hegedüs: Algorithms for covering polygons by rectangles seems to address a similar problem. And since this paper is from 1982, it might be interesting to look at the papers which cite this one. Additionally, this meeting seems to be discussing research problems related to this, so might be a starting point for keywords or names who do research in this idea.
I don't know if the computational geometry research has algorithms for your specific problem, or if these algorithms are easy/practical enough to implement. Here is how I would approach it if I had to do it without being able to look up previous work. This is just a direction, by far not a solution...
Formulate it as an optimization problem. You have discrete variables of which rectangles you choose (yes or no) and continuous variables (location and orientation of the triangles). Now you can set up two independent optimizations: a discrete optimization which picks the rectangles; and a continuous that optimizes for the location and orientation once rectangles are given. Interleave these two optimizations. Of course the difficulty lies in the formulation of optimizations, and designing your error energy such that it does not get stuck in some strange configurations (local minima). I'd try to get the continuous as a least squares problem such that I can use standard optimizations libraries.
I think this problem is suitable for solving with genetic algorithm and/or evolutionary strategy algorithm. I've done similar box packing problem with the help of evolutionary strategy algorithm of some kind. Check this out in my blog.
So if you will use such approach - encode into chromosomes box:
x coordinate
y coordinate
angle
Then try to minimize such fitness function-
y = w1 * box_intersection_area +
w2 * box_area_out_of_shape +
w3 * average_circle_radius_in_free_space
Choose weights w1,w2,w3 such as to affect importance of factors. When genetic algorithm will find partial solution - remove boxes which still overlaps together or are out of shape - and you will have at least legal (but not necessary optimal) solution.
good luck in this interesting problem !
It is NP hard indeed and since it has hi-tech application, reasonably efficients approximate strategies are not even in patents, let alone published papers.
The best you can do with a limited budget is to start by limiting the problem. Assume that all rectangles are exactly the same, Assume that all rectangles which are binary sub-divisions of your standard rectangle are also allowed since you can efficiently pre-pack them to fit your core division. For extra points you can also form several fixed schemas for gluing core rectangles to cover a few larger shapes with substantially different proportions. Assume that you can change dimensions of your standard rectangle/cell as long as the rest (pre-packing and gluing schema) remains the same - this gives you parameters to decide approximate size of the core rectangle based on rectangles you are given.
Now you can play with aspect ratios to approximate the error such limited system could guarantee. For the first iterations assume that it can have 50% error with a simple sub-division schema and then change schema to reduce the error but without increasing asymptotic complexity of pre-packing. At the end of the day you are always just assigning given rectangles to your pre-calculated and now fixed grid and binary sub-divisions - meaning you are not trying to do a layout or backtrack at all - you are always happy with the first approximate fit into the grid.
Work on defining classes of rectangles that pack well with your schema - that's again to keep the whole process inverted - you are never trying to actually fit what you are given - you are defining what you have to be given in order to fir it well - then you punt the rest as error since it is approximation.
Then you can try to do a bit more, but not much more - any slip into backtracking or nailing arbitrary small error and it's exponential.
If you are at a research facility and can get some supercomputer time - run a set of exhaustive searches with pathological mixes there just to see how optimal packing may look like and to see if you can derive a few more sub-division schemas and/or classes of rectangle sets.
That should be enough for the first 2 yrs or research :-)
I'm building a 2D physics engine and I want to add broad-phase collision detection, though I only know of 2 or 3 types:
Check everything against everything else (O(n^2) complexity)
Sweep and Prune (sort and sweep)
something about Binary Space Partition (not sure how to do this)
But surely there's more options right? what are they? And can either a basic description of each be provided or links to descriptions?
I've seen this but I'm asking for a list of algorithms available, not the best one for my needs.
In this case, "Broad phase collision detection" is a method used by physics engines to determine which bodies in their simulation are close enough to warrant further investigation and possibly collision resolution.
The best approach depends on the specific use, but the bottom line is that you want to subdivide your world space such that (a) every body is in exactly one subdivision, (b) every subdivision is large enough that a a body in a particular subdivision can only collide with bodies in that same subdivision or an adjacent subdivision, and (c) the number of bodies in a particular subdivision is as small as possible.
How you do that depends on how many bodies you have, how they're moving, what your performance requirements are, and how much time you want to spend on your engine. If you're talking about bodies moving around in a largely open space, the simplest technique would be divide the world into a grid where each cell is larger than your largest object, and track the list of objects in each cell. If you're building something on the scale of a classic arcade game, this solution may well suffice.
If you're dealing with bodies moving in a larger open world, a simple grid will become overwhelming pretty quickly, and you'll probably want some sort of a tree-based structure like quadtrees, as Arriu suggests.
If you're talking about moving bodies around within bounded spaces instead of open spaces, then you may consider a BSP tree; the tree partitions the world into 'space you can walk in' and 'walls', and clipping a body into the tree determines whether it's in a legal position. Depending on the world geometry, you can also use a BSP for your broad-phase detection of collisions between bodies in the world.
Another option for bodies moving in bounded space would be a portal engine; if your world can consist of convex polygonal regions where each side of the polygon is either a solid wall or a 'portal' to another concave space, you can easily determine whether a body is within a region with a point-in-polygon test and simplify collision detection by only looking at bodies in the same region or connected regions.
An alternative to QuadTrees or BSPTrees are SphereTrees (CircleTrees in 2D, the implementation would be more or less the same). The advantage that SphereTrees have are that they handle large loads of dynamic objects very well. If you're objects are constantly moving, BSPTrees and QuadTrees are much slower in their updates than a Sphere/Circle Tree would be.
If you have a good mix of static and dynamic objects, a reasonably good solution is to use a QuadTree/BSPTree for the statics and a Sphere/Cicle Tree for the dynamic objects. Just remember that for any given object, you would need to check it against both trees.
I recommend quadtree partitioning. It's pretty simple and it works really well. Here is a Flash demo of brute-force collision detection vs. quadtree collision detection. (You can tell it to show the quadtree structure.) Did you notice how quadtree collision detection is only 3% of brute force in that demo?
Also, if you are serious about your engine then I highly recommend you pick up real-time collision detection. It's not expensive and it's a really great book which covers everything you would ever want to know. (Including GPU based collision detection.)
All of the acceleration algorithms depend on using an inexpensive test to quickly rule out objects (or groups of objects) and thereby cut down on the number of expensive tests you have to do. I view the algorithms in categories, each of which has many variations.
Spatial partitioning. Carve up space and cheaply exclude candidates that are in different regions. For example, BSP, grids, octrees, etc.
Object partitioning. Similar to #1, but the clustering is focused on the objects themselves more than the space they reside in. For example, bounding volume hierarchies.
Sort and sweep methods. Put the objects in order spatially and rule out collisions among ones that aren't adjacent.
1 and 2 are often hierarchical, recursing into each partition as needed. With 3, you can optionally iterate along different dimensions.
Trade-offs depend a lot on scene geometry. Do objects cluster or are they evenly or sparsely distributed? Are they all about the same size or are there huge variations in size? Is the scene dynamic? Can you afford a lot of preprocessing time?
The "inexpensive" tests are actually along a spectrum of really-cheap to kind-of-expensive. Choosing the best algorithm means minimizing the ratio of the cost of the inexpensive testing to the reduction in the number of expensive tests. Beyond the algorithmic concerns, you get into tuning, like worrying about cache locality.
An alternative are plain grids, say 20x20 or 100x100 (depends on your world and memory size). The implementation is simpler than a recursive structure such as quad/bsp-trees (or sphere trees for that matter).
Objects crossing cell borders are a bit simpler in this case, and do not degenerate as much as an naive implementation of a bsp/quad/oct-tree might do.
Using that (or other techinques), you should be able to quickly cull many pairs and get a set of potential collisions that need further investigation.
I just came up with a solution that doesn't depend on grid size and is probably O(nlogn) (that is the optimum when there are no collisions) though worst at O(nnlogn) (when everything collides).
I also implemented and tested it, here is the link to the source. But I haven't compared it to the brute force solution.
A description of how it works:
(I'm looking for collisions of rectangles)
on x axis I sort the rectangles according to their right edge ( O(nlogn) )
for every rect in the sorted list I take the left edge and do a binary search until I find the rightmost edge at the left of the left edge and insert these rectangles between these indices in a possible_Collision_On_X_Axis set ( those are n rectangles, logn for the binary search, n inserts int the set at O(log n)per insert)
on y axis I do the same
in each of the sets I now have possible collisions on x (in one set) and on y(int the other), I intersect these sets and now I have the collisions on both the x axis and y axis (that means I take the common elements) (O(n))
sorry for the poor description, I hope you understand better from the source, also an example illustrated here: image
You might want to check out what Scott did in Chipmunk with spacial hashing. The source is freely available. I think he used a similar technique to Box-2D if not for collision, definitely for contact persistence.
I've used a quad-tree in a larger project, which is good for game objects that don't move much (less removals & re-insertions). The same principle applies for octrees.
The basic Idea Is, you create a recursive tree structure, which stores 4(for quad), or 8(oct) child objects of the same type as the tree root. Each node in the tree represents a section of Cartesian space, for example, -100 -> +100 on each applicable axis. each child represents that same space, but subdivided by half (an immediate child of the example would represent, for example, -100->0 on X axis, and -100->0 on Y axis).
Imagine a square, or plane, with a given set of dimensions. Now draw a line through the centre on each axis, dividing that plane into 4 smaller planes. Now take one of them and do It again, and again, until you reach a point when the size of the plane segment Is roughly the size of a game object. Now you have your tree. Only objects occupying the same plane, can possibly collide. When you have determined which objects can collide, You generate pairs of possible collisions from them. At this stage, broadphase Is complete, and you can move onto narrow phase collision detection, which Is where your more precise, and expensive calculations are.
The purpose of Broadphase, Is to use inexpensive calculations to generate possible collisions, and cull out contacts that cannot occur, thus reducing the work your narrow phase algorithm has to perform, In turn, making your entire collision detection algorithm more efficient.
So before you go ahead and attempt to implement such an algorithm, as yourself:
How many objects are in my game?
If there are a lot, I probably need a broadphase. If not, then the Nnarrowphase should suffice.
Also, am I dealing with many moving objects?
Tree structures generally are slowed down by moving objects, as they can change their position in the tree over time, simply by moving. This requires that objects be removed and reinserted each frame (potentially), which is less than Ideal.
If this is the case, you would be better off with Sweep and Prune, which maintains min/max heaps of the extents of the shapes in your world. Objects do not need to be reinserted, but the heaps need to be resorted each frame, thought this is generally faster than a tree wide, traversal with removals and reinsertions.
Depending on your coding experience, one may be more complicated to code over another. Personally I have found trees to be more intuitive to code, but less efficient, and more prone to error, since they raise other issues, such as what to do if you have an object that sits directly on top of an axis, or the centre of the root node. This can be solved by using loose trees, which have some spacial overlap, so that one child node is always prioritised over others when such a case occurs.
If the space that your objects move within is bounded, then you could use a grid to subdivide your objects. Put each object into every grid cell which the object covers (fully or partially). To check if object A collides with any other object, determine which grid cells object A covers, then get the list of unique objects in those cells, and finally test object A against each unique object. This approach works best if most objects are usually contained in a single grid cell.
If your space is not bounded, then you will need to implement some sort of dynamic grid that can grow on demand in each of the four directions (in 2D).
The advantage of this approach over more adaptive algorithsm (i.e. BSP, Quadtree, Circletree) is that the cells can be accessed in O(1) time (i.e. constant time) rather than O(log N) time (i.e. logarithmic time). However these latter algorithms are able to adapt themselves to large variability in the density of objects. The grid approach works best when object density is fairly constant across the space.
I would like to recommend the introductory reference to game physics from Ian Millington, Game Physics Engine Development. It has a great section on broad phase collision detection with sample code.